Statistical Methods in Medical Research 4ed

242 Experimental design
Similarly, FC provides a test of the null hypothesis HC, that all the b j ˆ 0,
irrespective of the values of the ai.
If the additive model (9.1) is not true, the Residual SSq will be inflated by
discrepancies between E…yij† and the approximations given by the best-fitting
additive model, and the Residual MSq will thus be an unbiased estimate of a
quantity greater than the random variance. How do we know whether this has
happened? There are two main approaches, the first of which is to examine
residuals. These are the individual expressions yij yi: y:j ‡ y. Their sum of
squares was obtained, from (9.4), by subtraction, but it could have been obtained
by direct evaluation of all the N residuals and by summing their squares. These
residuals add to zero along each row and down each column, like the discrepancies
between observed and expected frequencies in a contingency table (§8.6), and
(as for contingency tables) the number of DF, …r 1†…c 1†, is the number of
values of residuals which may be independently chosen (the others being then
automatically determined). Because of this lack of independence the residuals are
not quite the same as the random error terms eij of (9.2), but they have much
the same distributional properties. In particular, they should not exhibit any
striking patterns. Sometimes the residuals in certain parts of the two-way
table seem to have predominantly the same sign; provided the ordering of the
rows or columns has any meaning, this will suggest that the row-effect constants
are not the same for all columns. There may be a correlation between the size of
the residual and the `expected' value* yi: ‡ y:j y: this will suggest that a change
of scale would provide better agreement with the additive model.
A second approach is to provide replication of observations, and this is
discussed in more detail after Example 9.1.
Example 9.1
Table 9.3 shows the results of a randomized block experiment to compare the effects on
the clotting time of plasma of four different methods of treatment of the plasma. Samples
of plasma from eight subjects (the `blocks') were assigned in random order to the four
treatments.
The correction term, T 2 =N, denoted here by CT, is needed for three items in the SSq
column, and it is useful to calculate this at the outset. The analysis is straightforward, and
the F tests show that differences between subjects and treatments are both highly significant.
Differences between subjects do not interest us greatly as the main purpose of the
experiment was to study differences between treatments. The standard error of the
p
difference between two treatment means is ‰2…0 6559†=8Š ˆ 0 405. Clearly, treatments
1, 2 and 3 do not differ significantly among themselves, but treatment 4 gives a significantly
higher mean clotting time than the others.
*This is the value expected on the basis of the average row and column effects, as may be seen from the equivalent
expression y ‡…yi: y†‡…y:j y†